1. **State the problem:** Solve the differential equation $$(1 + x^2) \frac{dy}{dx} + xy = 0$$ with the initial condition $y(0) = 2$ using the method of separation of variables. 2. **Rewrite the equation:** Move terms to isolate $\frac{dy}{dx}$: $$ (1 + x^2) \frac{dy}{dx} = -xy $$ $$ \frac{dy}{dx} = -\frac{xy}{1 + x^2} $$ 3. **Separate variables:** Write the equation as $$ \frac{dy}{y} = -\frac{x}{1 + x^2} dx $$ 4. **Integrate both sides:** $$ \int \frac{1}{y} dy = - \int \frac{x}{1 + x^2} dx $$ 5. **Integrate left side:** $$ \int \frac{1}{y} dy = \ln|y| + C_1 $$ 6. **Integrate right side:** Use substitution $u = 1 + x^2$, so $du = 2x dx$, then $$ \int \frac{x}{1 + x^2} dx = \frac{1}{2} \int \frac{1}{u} du = \frac{1}{2} \ln|u| + C_2 = \frac{1}{2} \ln(1 + x^2) + C_2 $$ 7. **Combine results:** $$ \ln|y| = -\frac{1}{2} \ln(1 + x^2) + C $$ where $C = C_2 - C_1$ is a constant. 8. **Exponentiate both sides:** $$ |y| = e^C (1 + x^2)^{-1/2} $$ Let $A = e^C > 0$, so $$ y = \pm A (1 + x^2)^{-1/2} $$ 9. **Apply initial condition $y(0) = 2$:** $$ 2 = \pm A (1 + 0)^{-1/2} = \pm A $$ So $A = 2$ and choose positive sign. 10. **Final solution:** $$ \boxed{y = \frac{2}{\sqrt{1 + x^2}}} $$ This is the particular solution satisfying the initial condition.